62 lines
2.7 KiB
Text
62 lines
2.7 KiB
Text
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/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Jeremy Avigad
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Cardinality calculations for finite sets.
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-/
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import data.finset.comb
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open nat eq.ops
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namespace finset
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variable {A : Type}
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variable [deceq : decidable_eq A]
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include deceq
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theorem card_add_card (s₁ s₂ : finset A) : card s₁ + card s₂ = card (s₁ ∪ s₂) + card (s₁ ∩ s₂) :=
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finset.induction_on s₂
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(show card s₁ + card ∅ = card (s₁ ∪ ∅) + card (s₁ ∩ ∅),
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by rewrite [union_empty, card_empty, inter_empty])
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(take s₂ a,
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assume ans2: a ∉ s₂,
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assume IH : card s₁ + card s₂ = card (s₁ ∪ s₂) + card (s₁ ∩ s₂),
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show card s₁ + card (insert a s₂) = card (s₁ ∪ (insert a s₂)) + card (s₁ ∩ (insert a s₂)),
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from decidable.by_cases
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(assume as1 : a ∈ s₁,
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assert H : a ∉ s₁ ∩ s₂, from assume H', ans2 (mem_of_mem_inter_right H'),
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begin
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rewrite [card_insert_of_not_mem ans2, union.comm, -insert_union, union.comm],
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rewrite [insert_union, insert_eq_of_mem as1, insert_eq, inter.distrib_left, inter.comm],
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rewrite [singleton_inter_of_mem as1, -insert_eq, card_insert_of_not_mem H, -*add.assoc],
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rewrite IH
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end)
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(assume ans1 : a ∉ s₁,
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assert H : a ∉ s₁ ∪ s₂, from assume H',
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or.elim (mem_or_mem_of_mem_union H') (assume as1, ans1 as1) (assume as2, ans2 as2),
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begin
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rewrite [card_insert_of_not_mem ans2, union.comm, -insert_union, union.comm],
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rewrite [card_insert_of_not_mem H, insert_eq, inter.distrib_left, inter.comm],
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rewrite [singleton_inter_of_not_mem ans1, empty_union, add.right_comm],
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rewrite [-add.assoc, IH]
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end))
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theorem card_union (s₁ s₂ : finset A) : card (s₁ ∪ s₂) = card s₁ + card s₂ - card (s₁ ∩ s₂) :=
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calc
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card (s₁ ∪ s₂) = card (s₁ ∪ s₂) + card (s₁ ∩ s₂) - card (s₁ ∩ s₂) : add_sub_cancel
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... = card s₁ + card s₂ - card (s₁ ∩ s₂) : card_add_card
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theorem card_union_of_disjoint {s₁ s₂ : finset A} (H : disjoint s₁ s₂) :
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card (s₁ ∪ s₂) = card s₁ + card s₂ :=
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by rewrite [card_union, ↑disjoint at H, inter_empty_of_disjoint H]
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theorem card_le_card_of_subset {s₁ s₂ : finset A} (H : s₁ ⊆ s₂) : card s₁ ≤ card s₂ :=
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have H1 : disjoint s₁ (s₂ \ s₁),
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from disjoint.intro (take x, assume H1 H2, not_mem_of_mem_diff H2 H1),
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calc
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card s₂ = card (s₁ ∪ (s₂ \ s₁)) : union_diff_cancel H
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... = card s₁ + card (s₂ \ s₁) : card_union_of_disjoint H1
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... ≥ card s₁ : le_add_right
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end finset
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